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by Dr PHUA Kai Lit

School of Medicine and Health Sciences

Monash University Malaysia

Bandar Sunway, Malaysia

Read article on Racism and science

Click for Teach Yourself Statistics

Companion book: "Statistics Made Simple for Healthcare and Social Science Professionals and Students"

by Wong Kam Cheong and Phua Kai Lit (Serdang: UPM Press 2006)

To view this book

**POPULATION**: Universe of all units being studied. If we want to study lung cancer among Malaysians, then the study population will be all Malaysians. If we want to study lung cancer among Malaysian women, then the population will be all Malaysian women.

**SAMPLE**: Subset of the population

**RANDOM SAMPLE**: Each member of the population has an EQUAL

CHANCE of being chosen for the sample (simple random sample)

**SAMPLING METHODS** (#5 is nonrandom)

#1 Simple Random Sample

#2 Systematic Sample

Example:

Rank 100 people by age

Beginning with the 5th person, choose every tenth

i.e. Choose the 5th,15th,25th ... 85th,95th persons

#3 Stratified Sample: Composition of the sample reflects

composition of the population (proportional stratified sample)

Malaysian Population

Malays 60%

Chinese 25%

Indians 10%

Others 5%

Proportional Stratified Sample of 1000 Malaysians

Malays 600 (60%)

Chinese 250 (25%)

Indians 100 (10%)

Others 50 (5%)

#4 Cluster Sample

Divide population into groups

Random sample of groups is chosen

Count every unit in each and every group selected

Example:

Divide entire city into "city blocks"

Random sample of blocks is chosen

Count every person in each city block selected

#5 Nonrandom/Convenience Sample

Example:

Interview people in a shopping mall. This is nonrandom because not everyone in the population goes shopping at the mall. People who are wheelchair-bound are less likely to visit the mall. This is also the case with people who have problems walking. Also, the mall may be too far away for those who don't have cars. If the interview is done during a weekday, people who work are unlikely to be at the mall (housewives and retirees are more likely to be interviewed on weekdays).

**VALIDITY**: You are actually measuring what you want to measure. Example: if IQ really measures intelligence, then IQ is high in validity. If it does not actually measure intelligence, then it is low in validity.

**RELIABILITY**: Refers to stability of measurement. Example: A research instrument is high in reliability if it gives consistent readings when it is used on the same subject even if the subject is measured at different times. Example of a "research instrument" is a survey questionnaire.

**MEASURES OF CENTRAL TENDENCY** (used to summarise a dataset)

1. Arithmetic **Mean** - affected by extreme values

2. **Median**

3. **Mode**

**MEASURES OF DISPERSION** (used to determine how spread out a dataset is)

1. **Range** - difference between highest and lowest values

2. **Inter-quartile range** - difference between the first and third quartiles

3. **Variance**

4. **Standard Deviation** - the higher the Standard Deviation, the more spread out the data. The Standard Deviation is the square root of the Variance.

* For normally distributed data, use the mean and the standard deviation. For skewed data, use the median and the inter-quartile range.

**THE STANDARD DEVIATION IS A VERY IMPORTANT MEASURE** - Under a Standardised Normal Curve,

68.3% of the data are found +1 or -1 standard deviation from the mean

95.5% of the data are found +2 or -2 standard deviations from the mean

99.7% are found +3 or -3 standard deviations from the mean

**LEVEL OF MEASUREMENT OF DATA**

1. **Nominal data**: qualitative, categorical data. Example: ethnicity, gender, religion.

2. **Ordinal data**: Rank-ordered data. Data are grouped from low to high. But we cannot say how much lower or how much higher. Example: "low anxiety", "moderate anxiety" and "high anxiety".

3. **Interval data**: quantitative data with arbitrary zero. Example of interval data: temperature measured using the Celsius scale.

4. **Ratio level data**: Similar to Interval Data but in addition, it has an **absolute zero (true zero)** e.g. income, temperature measured using the Kelvin scale.

** NOTE: For Ratio Data, we can use ratio level, interval level, ordinal level and nominal level statistical tests.
For Interval Data, we can use interval level, ordinal level and nominal level tests.
For Ordinal Data, we can use ordinal level and nominal level tests.
But for Nominal Data, we can only use nominal level statistical tests.**

If we are doing research on a large population, we need not study each and every individual in the population. All we need to do is choose a sample (RANDOM and REPRESENTATIVE) from the population. We can use our findings from the sample to infer (draw conclusions) about the population.

**Research Hypothesis/Alternative Hypothesis**: the hypothesis we wish to confirm. Also called H-one and written as H_{1}

**Null Hypothesis**: opposite of the Research Hypothesis. Also called H-nought and written as H_{0}

Examples:

Research Hypothesis - there is a statistically significant association between X and Y

Null Hypothesis - there is no statistically significant association between X and Y

Any association seen is due to chance

Research Hypothesis - there is a statistically significant difference between the two population means

Null Hypothesis - there is no statistically significant difference between the two population means

Any difference seen is due to chance

**Significance Level/Confidence Level (denoted by Greek symbol alpha)**

Usually set at 0.05 or 0.01

An alpha of 0.05 means we wish to test the statement "the probability of what we see occuring by chance is less than 5%"

An alpha of 0.01 means we wish to test the statement "the probability of what we see occuring by chance is less than 1%"

(An alpha of 0.05 can also be interpreted as the probability of rejecting a true Null Hypothesis).

**Type 1 Error and Type 2 Error**

Type 1 Error (or alpha): rejecting a true Null Hypothesis

Type 2 Error (or beta) : accepting a false Null Hypothesis

**Power of a test**

Power = 1 - beta (i.e. 1 - probability of Type 2 Error)

Normally, the power of a test should be at least 80% or 0.8

Thus, the probability of detecting an effect is 80% and the probability of not detecting the effect is 20%

Power of a study can be raised by increasing the sample size.

**Interpretation of Statistical Output**

We reject the Null Hypothesis if the **test statistic e.g. the calculated chi-square or the calculated t** exceeds the **critical value**.

We accept (strictly: "fail to reject")the Null Hypothesis if the **test statistic e.g. the calculated chi-square or the calculated t ** does not
exceed the **critical value**

**p-value**: If p < 0.05, we conclude that there is less than a 5% probability that what we see has occurred by chance

If p < 0.01, we conclude that there is less than a 1% probability that what we see has occurred by chance

**Thus, the p-value is the probability that the observed relationship (e.g., between variables) or a difference (e.g., between means) in a sample occurred purely by chance, and that in the population from which the sample was drawn, no such relationship or differences actually exist.**

**One-tail test and Two-tail test**

Two-tail test: we test to see if there is a difference between X and Y

One-tail test: we test to see if there is a difference between X and Y **in one particular direction**

Example: We test to see if X > Y

Example: We test to see if X < Y

**Standardised Normal Distribution**: See description under "Standard Deviation"

In a normal distribution, the mean, median and mode are equal.

The curve is bell-shaped and symmetrical

The standardised normal distribution curve has a mean of zero and a standard deviation of 1.

**Degree of freedom**: the degree of freedom depends on the sample size or number of categories. The critical value of a statistical test changes with changes in the degree of freedom.

**COMMON STATISTICAL TESTS**

1. Chi-square test of goodness-of-fit, single sample (NOMINAL DATA). Degree of freedom is n-1 (where n = number of categories)

2. **Chi-square test of independence** (NOMINAL DATA). Degree of freedom is (r-1) X (c-1) where r = number of rows and c = number of columns in a **contingency table**.

To test if there is a statistically significant **association between two variables**.

3. **t-test for two independent samples** (INTERVAL DATA). Degree of freedom = (n_{1}-1) + (n_{2}-1)

To test if there is a statistically significant **difference between the two population means** from which the two samples are selected.

4. t-test for two matched samples (INTERVAL DATA). Degree of freedom is n-1 where n = the number of pairs

To test if there is a statistically significant **difference between the two population means** from the two matched samples.

5. If there are more than two samples, we use the ANOVA test (Analysis of Variance)

**NOTE: IF THERE ARE MORE THAN TWO SAMPLES, IT IS INCORRECT TO USE THE T-TEST TO MAKE PAIRWISE COMPARISONS**

Example: if there are 3 samples, it is incorrect to compare mean #1 with mean #2, mean #1 with mean #3, mean #2 with mean #3. The ANOVA test should be done on the three means instead.

**CHOOSING A TEST**

1. What is the level of measurement? Nominal, ordinal or interval?

2. How many samples? One, two or more?

3. If two samples, are they independent or paired/matched?

4. Choose the test. **Make sure the assumptions of the test are not violated**

**ASSUMPTIONS OF CHI-SQUARE TEST OF INDEPENDENCE**

1. Nominal data (ordinal data also OK)

2. 25 =< n =<250 (preferably)

3. Random sample

4. Expected value of each cell is at least 5 (if not, you should combine some of the categories)

**INTERPRETING RESULTS OF CHI-SQUARE TEST**

**H _{0} is "There is no association between X and Y. Any association seen is due to chance alone"
H_{1} is "There is a statistically significant association between X and Y"**

**
Reject H _{0} if the calculated chi-square exceeds the critical value
Reject H_{0} if p is less than 0.05 (if testing at alpha = 0.05)**

**ASSUMPTIONS OF T-TEST FOR TWO INDEPENDENT SAMPLES**

1. Random

2. Interval data

3. Normal distribution in both groups

4. Preferably n < 30 (for each sample).

**INTERPRETING RESULTS OF T-TEST**

**H _{0} is "There is no difference between population mean X and population mean Y. Any difference seen is due to chance alone"
H_{1} is "There is a difference between population mean X and population mean Y"**

**
Reject H _{0}if the calculated t statistic exceeds the critical value
Reject H_{0} if p is less than 0.05 (if testing at alpha = 0.05)**

** IMPORTANT: What is STATISTICALLY SIGNIFICANT may not be CLINICALLY SIGNIFICANT.**

**Correlation**: a measure of how two variables go together.

**Pearson's r** (also called Pearson's correlation coefficient) is a measure of **linear** relationship between two variables.

A value of +1 means a perfect positive linear relationship.

A value of -1 means a perfect negative linear relationship.

A value of 0 means no linear relationship.

**Assumptions for using Pearson's r**:

Randomness

Linear relationship exists

Both variables are normally distributed

Variables measured at Interval level

**It is incorrect to use r for variables measured at nominal or ordinal level**

**Correlations can also be nonlinear. For nonlinear correlations, we do not use Pearson's r but some other correlation coefficient e.g. Spearman's correlation coefficient.**

**NOTE: CORRELATION DOES NOT IMPLY CAUSATION**

Just because two variables are correlated does not necessarily mean that one causes the other.

**Regression**: used to predict how independent variables (X1, X2 etc) affect a dependent variable (Y).

**Simple Regression**: Has only 1 dependent variable Y and 1 independent variable X

**Multiple Regression**: Has 1 dependent variable Y but two or more
independent variables X1, X2 etc

Example

Simple regression - predict INCOME (Y) from YEARS OF EDUCATION (X)

Multiple regression - predict INCOME (Y) from YEARS OF EDUCATION (X1) and YEARS OF WORKING EXPERIENCE (X2)

(Variables measured at the nominal level such as "Gender" can also be used as independent variables in regression. They are used as "dummy variables").

**r ^{2}**: An indicator of the "amount of variance of the dependent variable accounted for" by the regression equation. Also called the

**regression coefficient**: If we have a regression equation
Y = 0.3X1 + 4X2, then the regression coefficient of X1 is 0.3 and the regression coefficient of X2 is 4.

This means that when X1 increases by 1 unit, Y will increase by 0.3

Also, when X2 increases by 1 unit, Y will increase by 4 units.

**Example**

Diastolic blood pressure of sample of men aged 30-50 are plotted against their age

Y = 40 + 1.5X

(Y = diastolic blood pressure, X = age)

Interpretation: For these men, each year of increase in age raises the diastolic bp by 1.5 mm Hg

If man is 50 years old, the predicted diastolic bp is

40 + 1.5(50) = 115 mm Hg

NOTE: It is **incorrect to extrapolate in regression analysis** i.e. if your sample consists of men aged 30 - 50, you should not use the regression model to predict the blood pressure of men whose ages are below 30 or above 50

**WHAT TO LOOK FOR IN A GOOD REGRESSION MODEL**

1. Are the dependent variable and independent variables properly "operationalised" (defined and measured)?

2. Do the relationships between the dependent variable and independent variables make sense?

3. Is the relationship between the dependent variable and each independent variable linear in nature? Do **scatterplots**

4. Examine the r^{2}. The higher the r^{2}, the "better" the model i.e. more of the variance in the
dependent variable is accounted for.

5. Examine the sign of each regression coefficient. Do they make sense?

6. Check if each of the regression coefficients are statistically significant (p < 0.05 or p < 0.01)

7. Check if the following problems exist:

Multicollinearity (the VIF or Variance Inflation Factor should not be
greater than 10. Ideally, the VIF should be less than 5)

Autocorrelation of error term (serial correlation of error term) (use the Durbin-Watson test)

Heteroscedasticity (use either the Breusch-Pagan test or the NCV test i.e. Non-Constant Variance score test).

8. If there are more than one regression model, compare and contrast between them

Durbin-Watson test (d):

If d < dL, reject H_{0}: ρ = 0

If d > dU, do not reject H_{0}: ρ = 0 i.e. we conclude that serial correlation is not a problem

If dL< d < dU, test is inconclusive

Breusch-Pagan and NCV tests:

If unable to reject H_{0} (p>.05), then we can conclude that there is no problem with heteroscedasticity.

**Confidence Interval**: The interval within which something is likely to be found

A 95% Confidence Interval for the population mean indicates (loosely-speaking) that there is a 95% probability that the population mean actually lies within that particular Confidence Interval. Strictly speaking, it means that if you take 100 samples and calculate the sample means 100 times, 95 of these will fall within the 95% confidence interval.

**Skew**: If a curve is slanted to the right, it is skewed to the **left**.

If a curve is slanted to the left, it is skewed to the **right**

**Nonparametric Tests**: Statistical tests which make no assumptions about the parent distribution.

**Parametric Tests**: Statistical tests which assume that the population distribution has a particular form e.g. a normal distribution. The t-test is a parametric test as it assumes normal distribution.

**Standard Error of the Mean**: We take samples from a population.
For each sample, we calculate its mean. We then plot the sample means and we will get a curve (called "the sampling distribution of the means"). The curve will have a standard deviation. This standard deviation is the **standard error of the mean**. It is used to calculate confidence intervals.

The smaller the standard error of the mean, the more closely the sample mean estimates the true population mean.

**Odds Ratio**: This is commonly used in public health research nowadays.

** References**

Cassens, B.J. 1992 "Preventive Medicine and Public Health" 2nd ed. Philadelphia: Harwal Publishing

Champion, D.J. 1981 "Basic Statistics for Social Research" 2nd ed. New York: MacMillan

Porkess, R. 1991 "The Harper Collins Dictionary of Statistics" New York: HarperPerennial

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